Vector & 3D Geometry for Competitive Exam Preparation

Vector & 3D Geometry

3D Lines 3D Planes Scalar Product Vector Algebra Basics Vector Product

Q. If A = 2i - j and B = -i + 3j, what is the value of A · B?

A. -1
B. 1
C. 5
D. 7

Solution

A · B = (2)(-1) + (-1)(3) = -2 - 3 = -5.

Correct Answer: A — -1

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Q. If A = 6i + 8j and B = 3i + 4j, what is the scalar product A · B?

A. 50
B. 54
C. 60
D. 66

Solution

A · B = (6)(3) + (8)(4) = 18 + 32 = 50.

Correct Answer: B — 54

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Q. If A = 7i + 24j and B = 24i - 7j, calculate A · B.

A. 0
B. 1
C. 2
D. 3

Solution

A · B = (7)(24) + (24)(-7) = 168 - 168 = 0.

Correct Answer: A — 0

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Q. If A = i + 2j + 3k and B = 4i + 5j + 6k, find A · B.

A. 32
B. 30
C. 28
D. 34

Solution

A · B = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32.

Correct Answer: B — 30

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Q. If A(1, 2, 3) and B(4, 5, 6) are two points in space, what is the vector AB?

A. (3, 3, 3)
B. (2, 3, 4)
C. (1, 1, 1)
D. (0, 0, 0)

Solution

Vector AB = B - A = (4-1, 5-2, 6-3) = (3, 3, 3).

Correct Answer: A — (3, 3, 3)

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Q. If A(1, 2, 3) and B(4, 5, 6) are two points, what is the vector AB?

A. (3, 3, 3)
B. (3, 3, 0)
C. (0, 0, 0)
D. (1, 1, 1)

Solution

Vector AB = B - A = (4-1, 5-2, 6-3) = (3, 3, 3).

Correct Answer: A — (3, 3, 3)

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Q. If A(2, 3, 4) and B(1, 0, -1) are two points in space, find the vector AB.

A. (1, 3, 5)
B. (1, -3, -5)
C. (1, 3, -5)
D. (1, -3, 5)

Solution

AB = B - A = (1 - 2, 0 - 3, -1 - 4) = (-1, -3, -5) = (1, 3, 5) in the opposite direction.

Correct Answer: A — (1, 3, 5)

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Q. If C = (0, 1, 2) and D = (3, 4, 5), what is C · D?

A. 10
B. 11
C. 12
D. 13

Solution

C · D = 0*3 + 1*4 + 2*5 = 0 + 4 + 10 = 14.

Correct Answer: A — 10

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Q. If C = (3, -1, 2) and D = (0, 4, -1), what is the value of C · D?

A. -2
B. 1
C. 2
D. 10

Solution

C · D = 3*0 + (-1)*4 + 2*(-1) = 0 - 4 - 2 = -6.

Correct Answer: A — -2

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Q. If C = (3, -1, 2) and D = (0, 4, -3), what is the value of C · D?

A. -10
B. 10
C. 0
D. 6

Solution

C · D = 3*0 + (-1)*4 + 2*(-3) = 0 - 4 - 6 = -10.

Correct Answer: A — -10

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Q. If C = (3, -1, 2) and D = (4, 2, 1), calculate C · D.

A. 10
B. 11
C. 12
D. 13

Solution

C · D = 3*4 + (-1)*2 + 2*1 = 12 - 2 + 2 = 12.

Correct Answer: B — 11

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Q. If C = (3, -2, 1) and D = (0, 4, -3), what is the value of C · D?

A. -10
B. 10
C. 0
D. 6

Solution

C · D = 3*0 + (-2)*4 + 1*(-3) = 0 - 8 - 3 = -11.

Correct Answer: A — -10

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Q. If E = (1, 1, 1) and F = (1, 2, 3), find E · F.

A. 1
B. 2
C. 3
D. 6

Solution

E · F = 1*1 + 1*2 + 1*3 = 1 + 2 + 3 = 6.

Correct Answer: D — 6

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Q. If E = (1, 1, 1) and F = (2, 2, 2), what is the scalar product E · F?

A. 3
B. 6
C. 9
D. 12

Solution

E · F = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.

Correct Answer: B — 6

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Q. If E = (2, -1, 3) and F = (1, 2, 0), what is E · F?

A. -1
B. 0
C. 1
D. 2

Solution

E · F = 2*1 + (-1)*2 + 3*0 = 2 - 2 + 0 = 0.

Correct Answer: A — -1

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Q. If E = (5, 5, 5) and F = (1, 2, 3), what is the scalar product E · F?

A. 30
B. 25
C. 20
D. 15

Solution

E · F = 5*1 + 5*2 + 5*3 = 5 + 10 + 15 = 30.

Correct Answer: A — 30

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Q. If E = (x, y, z) and F = (2, 3, 4) such that E · F = 10, what is the equation relating x, y, z?

A. 2x + 3y + 4z = 10
B. x + y + z = 10
C. x + 2y + 3z = 10
D. 2x + 3y + z = 10

Solution

E · F = x*2 + y*3 + z*4 = 10 gives the equation 2x + 3y + 4z = 10.

Correct Answer: A — 2x + 3y + 4z = 10

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Q. If G = (1, 0, 1) and H = (0, 1, 0), find G · H.

A. 0
B. 1
C. 2
D. 3

Solution

G · H = 1*0 + 0*1 + 1*0 = 0 + 0 + 0 = 0.

Correct Answer: A — 0

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Q. If G = (1, 1, 1) and H = (1, -1, 1), what is G · H?

A. 0
B. 1
C. 2
D. -1

Solution

G · H = 1*1 + 1*(-1) + 1*1 = 1 - 1 + 1 = 1.

Correct Answer: A — 0

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Q. If G = (1, 1, 1) and H = (2, 2, 2), what is G · H?

A. 3
B. 4
C. 5
D. 6

Solution

G · H = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.

Correct Answer: A — 3

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Q. If G = (2, 2) and H = (3, -1), what is G · H?

A. -1
B. 1
C. 5
D. 7

Solution

G · H = 2*3 + 2*(-1) = 6 - 2 = 4.

Correct Answer: C — 5

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Q. If I = (1, 1, 1) and J = (2, 2, 2), what is the scalar product I · J?

A. 3
B. 4
C. 5
D. 6

Solution

I · J = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.

Correct Answer: D — 6

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Q. If I = (1, 2, 3) and J = (3, 2, 1), what is I · J?

A. 10
B. 11
C. 12
D. 13

Solution

I · J = 1*3 + 2*2 + 3*1 = 3 + 4 + 3 = 10.

Correct Answer: A — 10

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Q. If I = (1, 2, 3) and J = (4, 5, 6), calculate I · J.

A. 32
B. 30
C. 28
D. 26

Solution

I · J = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.

Correct Answer: B — 30

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Q. If I = (a, b, c) and J = (2, 2, 2) such that I · J = 12, what is the relationship between a, b, c?

A. a + b + c = 6
B. 2a + 2b + 2c = 12
C. a + b + c = 12
D. 2a + 2b + 2c = 6

Solution

I · J = 2a + 2b + 2c = 12 gives the equation 2a + 2b + 2c = 12.

Correct Answer: B — 2a + 2b + 2c = 12

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Q. If M = (1, 2, 3) and N = (4, 5, 6), what is the scalar product M · N?

A. 32
B. 33
C. 34
D. 35

Solution

M · N = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.

Correct Answer: A — 32

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Q. If M = (2, 2, 2) and N = (3, 3, 3), what is M · N?

A. 12
B. 18
C. 24
D. 30

Solution

M · N = 2*3 + 2*3 + 2*3 = 6 + 6 + 6 = 18.

Correct Answer: A — 12

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Q. If the angle between two vectors A and B is 90 degrees, what is the value of A · B?

A. 1
B. 0
C. undefined
D. 1/2

Solution

If the angle is 90 degrees, A · B = |A||B|cos(90) = 0.

Correct Answer: B — 0

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Q. If the angle between vectors A = 2i + 3j and B = 4i + 5j is 60 degrees, find A · B.

A. 20
B. 25
C. 30
D. 35

Solution

A · B = |A||B|cos(60°) = √(2^2 + 3^2) * √(4^2 + 5^2) * 1/2 = √13 * √41 * 1/2 = 20.

Correct Answer: B — 25

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Q. If the position vector of a point is (5, 12), what is its distance from the origin?

A. 13
B. 12
C. 11
D. 10

Solution

Distance = √(5^2 + 12^2) = √(25 + 144) = √169 = 13

Correct Answer: A — 13

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Vector & 3D Geometry

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